Show that $\frac{G \oplus G \oplus G }{ (1,0,-1)G + (0,1,-1)G} = G$

Question

Let $G$ be an arbitrary abelian group and consider a subgroup of $P = G \oplus G \oplus G$ given by $$ S = (1,0,-1)G+(0,1,-1)G:=\left\{ (g_1,g_2,-g_1-g_2) \mid g_1,g_2 \in G \right\}. $$ Is it true that $P / S = G$? If it is true, what is a standard way of showing such equalities?

Answer 1

Let $\phi:G\oplus G\oplus G\to G$ defined by $\phi(x,y,z)=x+y+z$. Then $\phi$ is a surjective group homomorphism and $\ker\phi=\dots$. Now apply the fundamental theorem of isomorphism for groups and find that $P/S\simeq G$.